The strength of the rainbow Ramsey Theorem

Csima, Barbara F.; Mileti, Joseph R.

doi:10.2178/jsl/1254748693

Cited by 26 publications

(75 citation statements)

References 15 publications

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“…Then any function f : [ω] n → k has a homogenous set H. Theorem 2 (RRT k n ) Given n, k ∈ ω and a k-bounded function f : [ω] n → ω, there exists a rainbow set R for f. [3] proved that RRT 2 2 is strictly weaker than RT 2 2 under RCA 0 and RRT 2 2 is incomparable with WKL 0 over RCA 0 in 2009. Recently, Wang [8] showed that RRT 3 2 implies neither WKL 0 nor RRT 4 2 under RCA 0 . Theorem 3 (FS(k)) For each function f : [ω] k → ω and k ∈ ω, there is a free set S for f.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial principles between RRT 2 2 and RT 2 2

Kang

2014

Front. Math. China

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We study the strength of some combinatorial principles weaker than Ramsey theorem for pairs over RCA 0 . First, we prove that Rainbow Ramsey theorem for pairs does not imply Thin Set theorem for pairs. Furthermore, we get some other related results on reverse mathematics using the same method. For instance, Rainbow Ramsey theorem for pairs is strictly weaker than Erdős-Moser theorem under RCA 0 .

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Section: Introductionmentioning

confidence: 99%

Combinatorial principles between RRT 2 2 and RT 2 2

Kang

2014

Front. Math. China

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“…More recently, Csima and Mileti [3] studied the effectivity of the Rainbow Ramsey Theorem (for n tuples and k−bounded colorings). Fix numbers n, k. We say that a coloring of n−tuples c : 2 → ω is a recursive k-bounded coloring of pairs and X is a 2-random real, then X computes a rainbow for c. For more details, consult [3] or the discussion in Section 2.…”

Section: Effective Infinitary Combinatoricsmentioning

confidence: 99%

“…The Csima-Mileti Argument in RCA 0 + 2-RAN. We assume that the reader is familiar with [3]. Our main goal in this section is to sketch a proof, in the formal system RCA 0 + 2-RAN, of the theorem of Csima and Mileti [3, Section 3] that says for any recursive 2-bounded coloring of pairs c : [N] 2 → N and any 2-random real X , there is a rainbow R ⊆ N for c that is recursive in X .…”

Section: θ(|τ| τ) and (∀W < |τ|)¬θ(W τ W)mentioning

confidence: 99%

“…. are in the proof [3,Proposition 3.9]. By recursively and uniformly approximating the method of Csima and Mileti described in [3, pages 1316-1318] we can construct, in the formal system RCA 0 +2-RAN, a {0, 1}-valued recursive approximation function t (n, σ, s), σ ∈ 2 <N , n, s ∈ N, such that:…”

Section: θ(|τ| τ) and (∀W < |τ|)¬θ(W τ W)mentioning

confidence: 99%

“…and Mileti give in [3,Proposition 3.9]). Now, the axiom 2-RAN says that there exists a real X ∈ 2 N such that X ∈ [T n ], for some n ∈ N, and via the same decoding procedure described in [3,Proposition 3.9] there is an X -recursive rainbow R for c. Thus, we have essentially used the CsimaMileti method [3,Section 3] to prove the following theorem.…”

Section: This Is Essentially the Argument That Csimamentioning

confidence: 99%

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Random reals, the rainbow Ramsey theorem, and arithmetic conservation

Conidis

Slaman

2013

J. symb. log.

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Abstract. We investigate the question "To what extent can random reals be used as a tool to establish number theoretic facts?" Let 2-RAN be the principle that for every real X there is a real R which is 2-random relative to X . In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory RCA 0 and so RCA 0 + 2-RAN implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem is not conservative over RCA 0 for arithmetic sentences. Thus, from the CsimaMileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that 2-RAN has non-trivial arithmetic consequences. In Section 4, we show that 2-RAN is conservative over1 -sentences. Thus, the set of first-order consequences of 2-RAN is strictly stronger than P − + I Σ 1 and no stronger than P − + B Σ 2 . §1. Introduction. One of the benefits to having a precise formulation of the concept "random infinite binary sequence" is that one can ask and precisely answer questions about what types of objects can be computed from random input and about what sorts of theorems can be proven from the existence of a random sequence.In [3], Csima and Mileti gave an intriguing example of the first type. Their example concerns the Rainbow Ramsey Theorem, which states that if C is a coloring of size-k subsets of N such that there is a uniform finite bound on the number of sets assigned to any particular color, then there is an infinite set X such that C is injective on the size-k subsets of X , i.e. X is a C -rainbow. As is described below, they give a proof of the Rainbow Ramsey Theorem for pairs (k = 2) by showing that if R is a sufficiently random sequence relative to C , then R can be used to compute a C -rainbow. In a sentence, Csima and Mileti show how a random source can be used to produce a solution to a infinitary-combinatorial problem. Further, since they also show that there is a recursive such C with no recursive rainbow, any general method to produce rainbows for colorings must be driven by some such non-recursive data.In this article, we analyze the second question with respect to theorems about finite sets. For this question, we shift the setting from the examination of the computation

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Iterative Forcing and Hyperimmunity in Reverse Mathematics

Patey

2015

Lecture Notes in Computer Science

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Abstract. The separation between two theorems in reverse mathematics is usually done by constructing a Turing ideal satisfying a theorem P and avoiding the solutions to a fixed instance of a theorem Q. Lerman, Solomon and Towsner introduced a forcing technique for iterating a computable non-reducibility in order to separate theorems over omega-models. In this paper, we present a modularized version of their framework in terms of preservation of hyperimmunity and show that it is powerful enough to obtain the same separations results as Wang did with his notion of preservation of definitions.

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The strength of the rainbow Ramsey Theorem

Cited by 26 publications

References 15 publications

Combinatorial principles between RRT 2 2 and RT 2 2

Combinatorial principles between RRT 2 2 and RT 2 2

Random reals, the rainbow Ramsey theorem, and arithmetic conservation

Iterative Forcing and Hyperimmunity in Reverse Mathematics

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